ECE471-571 – Pattern Recognition Lecture 4 – Parametric Estimation Hairong Qi, Gonzalez Family Professor Electrical Engineering and Computer Science University of Tennessee, Knoxville http://www.eecs.utk.edu/faculty/qi Email: hqi@utk.edu

Pattern Classification Statistical Approach Non-Statistical Approach Supervised Unsupervised Decision-tree Basic concepts: Baysian decision rule (MPP, LR, Discri.) Basic concepts: Distance Agglomerative method Syntactic approach Parameter estimate (ML, BL) k-means Non-Parametric learning (kNN) Winner-takes-all LDF (Perceptron) Kohonen maps NN (BP) Mean-shift Support Vector Machine Deep Learning (DL) Dimensionality Reduction FLD, PCA Performance Evaluation ROC curve (TP, TN, FN, FP) cross validation Stochastic Methods local opt (GD) global opt (SA, GA) Classifier Fusion majority voting NB, BKS

Bayes Decision Rule Maximum Posterior Probability Likelihood Ratio If , then x belongs to class 1, otherwise, 2. Discriminant Function Case 1: Minimum Euclidean Distance (Linear Machine), Si=s2I Case 2: Minimum Mahalanobis Distance (Linear Machine), Si = S Case 3: Quadratic classifier , Si = arbitrary

Multivariate Normal Density

Estimating Normal Densities Calculate m, S

Covariance

*Why? – Maximum Likelihood Estimation Compare “likelihood”

*Derivation

* Derivative of a Quadratic Form

**Why? – Baysian Estimation Maximum likelihood estimation The parameters are fixed Find value for q that best agrees with or supports the actually observed training samples – likelihood of q w.r.t. the set of samples Baysian estimation Treat parameters as random variable themselves

** The pdf of the parameter (m) is Gaussian

** Derivation

** mn and sn Behavior of Bayesian learning Our best guess for m after observing n samples Measures our uncertainty about this guess Behavior of Bayesian learning The larger the n, the smaller the sn – each additional observation decreases our uncertainty about the true value of m As n approaches infinity, p(m|D) becomes more and more sharply peaked, approaching a Dirac delta function. mn is a linear combination between the sample mean and m0